Tangent¶

Tangent Basis¶

template <typename DerivedP0, typename DerivedP1> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_point_tangent_basis( const Eigen::MatrixBase<DerivedP0>& p0, const Eigen::MatrixBase<DerivedP1>& p1);¶

Compute a basis for the space tangent to the point-point pair.

Parameters:¶

const Eigen::MatrixBase<DerivedP0> &p0¶: First point
const Eigen::MatrixBase<DerivedP1> &p1¶: Second point

Returns:¶

A 3x2 matrix whose columns are the basis vectors.

template <typename DerivedP, typename DerivedE0, typename DerivedE1> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_edge_tangent_basis( const Eigen::MatrixBase<DerivedP>& p, const Eigen::MatrixBase<DerivedE0>& e0, const Eigen::MatrixBase<DerivedE1>& e1);¶

Compute a basis for the space tangent to the point-edge pair.

Parameters:¶

const Eigen::MatrixBase<DerivedP> &p¶: Point
const Eigen::MatrixBase<DerivedE0> &e0¶: First edge point
const Eigen::MatrixBase<DerivedE1> &e1¶: Second edge point

Returns:¶

A 3x2 matrix whose columns are the basis vectors.

template <typename DerivedEA0, typename DerivedEA1, typename DerivedEB0, typename DerivedEB1> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::edge_edge_tangent_basis( const Eigen::MatrixBase<DerivedEA0>& ea0, const Eigen::MatrixBase<DerivedEA1>& ea1, const Eigen::MatrixBase<DerivedEB0>& eb0, const Eigen::MatrixBase<DerivedEB1>& eb1);¶

Compute a basis for the space tangent to the edge-edge pair.

Parameters:¶

const Eigen::MatrixBase<DerivedEA0> &ea0¶: First point of the first edge
const Eigen::MatrixBase<DerivedEA1> &ea1¶: Second point of the first edge
const Eigen::MatrixBase<DerivedEB0> &eb0¶: First point of the second edge
const Eigen::MatrixBase<DerivedEB1> &eb1¶: Second point of the second edge

Returns:¶

A 3x2 matrix whose columns are the basis vectors.

template <typename DerivedP, typename DerivedT0, typename DerivedT1, typename DerivedT2> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_triangle_tangent_basis( const Eigen::MatrixBase<DerivedP>& p, const Eigen::MatrixBase<DerivedT0>& t0, const Eigen::MatrixBase<DerivedT1>& t1, const Eigen::MatrixBase<DerivedT2>& t2);¶

Compute a basis for the space tangent to the point-triangle pair.

\[ \begin{bmatrix} \frac{t_1 - t_0}{\|t_1 - t_0\|} & \frac{((t_1 - t_0)\times(t_2 - t_0)) \times(t_1 - t_0)}{\|((t_1 - t_0)\times(t_2 - t_0))\times(t_1 - t_0)\|} \end{bmatrix} \]

Parameters:¶

const Eigen::MatrixBase<DerivedP> &p¶: Point
const Eigen::MatrixBase<DerivedT0> &t0¶: Triangle’s first vertex
const Eigen::MatrixBase<DerivedT1> &t1¶: Triangle’s second vertex
const Eigen::MatrixBase<DerivedT2> &t2¶: Triangle’s third vertex

Returns:¶

A 3x2 matrix whose columns are the basis vectors.

Tangent Basis Jacobians¶

template <typename DerivedP0, typename DerivedP1> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_point_tangent_basis_jacobian( const Eigen::MatrixBase<DerivedP0>& p0, const Eigen::MatrixBase<DerivedP1>& p1);¶

Compute the Jacobian of the tangent basis for the point-point pair.

Parameters:¶

const Eigen::MatrixBase<DerivedP0> &p0¶: First point
const Eigen::MatrixBase<DerivedP1> &p1¶: Second point

Returns:¶

A (3*2)x6 matrix whose columns are the basis vectors.

template <typename DerivedP, typename DerivedE0, typename DerivedE1> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_edge_tangent_basis_jacobian( const Eigen::MatrixBase<DerivedP>& p, const Eigen::MatrixBase<DerivedE0>& e0, const Eigen::MatrixBase<DerivedE1>& e1);¶

Compute the Jacobian of the tangent basis for the point-edge pair.

Parameters:¶

const Eigen::MatrixBase<DerivedP> &p¶: Point
const Eigen::MatrixBase<DerivedE0> &e0¶: First edge point
const Eigen::MatrixBase<DerivedE1> &e1¶: Second edge point

Returns:¶

A (3*2)x9 matrix whose columns are the basis vectors.

template <typename DerivedEA0, typename DerivedEA1, typename DerivedEB0, typename DerivedEB1> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::edge_edge_tangent_basis_jacobian( const Eigen::MatrixBase<DerivedEA0>& ea0, const Eigen::MatrixBase<DerivedEA1>& ea1, const Eigen::MatrixBase<DerivedEB0>& eb0, const Eigen::MatrixBase<DerivedEB1>& eb1);¶

Compute the Jacobian of the tangent basis for the edge-edge pair.

Parameters:¶

const Eigen::MatrixBase<DerivedEA0> &ea0¶: First point of the first edge
const Eigen::MatrixBase<DerivedEA1> &ea1¶: Second point of the first edge
const Eigen::MatrixBase<DerivedEB0> &eb0¶: First point of the second edge
const Eigen::MatrixBase<DerivedEB1> &eb1¶: Second point of the second edge

Returns:¶

A (3*2)x12 matrix whose columns are the basis vectors.

template <typename DerivedP, typename DerivedT0, typename DerivedT1, typename DerivedT2> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_triangle_tangent_basis_jacobian( const Eigen::MatrixBase<DerivedP>& p, const Eigen::MatrixBase<DerivedT0>& t0, const Eigen::MatrixBase<DerivedT1>& t1, const Eigen::MatrixBase<DerivedT2>& t2);¶

Compute the Jacobian of the tangent basis for the point-triangle pair.

Parameters:¶

const Eigen::MatrixBase<DerivedP> &p¶: Point
const Eigen::MatrixBase<DerivedT0> &t0¶: Triangle’s first vertex
const Eigen::MatrixBase<DerivedT1> &t1¶: Triangle’s second vertex
const Eigen::MatrixBase<DerivedT2> &t2¶: Triangle’s third vertex

Returns:¶

A (3*2)x12 matrix whose columns are the basis vectors.

Relative Velocity¶

template <typename DerivedDP0, typename DerivedDP1> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_point_relative_velocity( const Eigen::MatrixBase<DerivedDP0>& dp0, const Eigen::MatrixBase<DerivedDP1>& dp1);¶

Compute the relative velocity of two points.

Parameters:¶

const Eigen::MatrixBase<DerivedDP0> &dp0¶: Velocity of the first point
const Eigen::MatrixBase<DerivedDP1> &dp1¶: Velocity of the second point

Returns:¶

The relative velocity of the two points

template <typename DerivedDP, typename DerivedDE0, typename DerivedDE1> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_edge_relative_velocity( const Eigen::MatrixBase<DerivedDP>& dp, const Eigen::MatrixBase<DerivedDE0>& de0, const Eigen::MatrixBase<DerivedDE1>& de1, const typename DerivedDP::Scalar alpha);¶

Compute the relative velocity of a point and an edge.

Parameters:¶

const Eigen::MatrixBase<DerivedDP> &dp¶: Velocity of the point
const Eigen::MatrixBase<DerivedDE0> &de0¶: Velocity of the first endpoint of the edge
const Eigen::MatrixBase<DerivedDE1> &de1¶: Velocity of the second endpoint of the edge
const typename DerivedDP::Scalar alpha¶: Parametric coordinate of the closest point on the edge

Returns:¶

The relative velocity of the point and the edge

template <typename DerivedDEA0, typename DerivedDEA1, typename DerivedDEB0, typename DerivedDEB1, typename DerivedCoords> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::edge_edge_relative_velocity( const Eigen::MatrixBase<DerivedDEA0>& dea0, const Eigen::MatrixBase<DerivedDEA1>& dea1, const Eigen::MatrixBase<DerivedDEB0>& deb0, const Eigen::MatrixBase<DerivedDEB1>& deb1, const Eigen::MatrixBase<DerivedCoords>& coords);¶

Compute the relative velocity of the edges.

Parameters:¶

const Eigen::MatrixBase<DerivedDEA0> &dea0¶: Velocity of the first endpoint of the first edge
const Eigen::MatrixBase<DerivedDEA1> &dea1¶: Velocity of the second endpoint of the first edge
const Eigen::MatrixBase<DerivedDEB0> &deb0¶: Velocity of the first endpoint of the second edge
const Eigen::MatrixBase<DerivedDEB1> &deb1¶: Velocity of the second endpoint of the second edge
const Eigen::MatrixBase<DerivedCoords> &coords¶: Two parametric coordinates of the closest points on the edges

Returns:¶

The relative velocity of the edges

template <typename DerivedDP, typename DerivedDT0, typename DerivedDT1, typename DerivedDT2, typename DerivedCoords> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_triangle_relative_velocity( const Eigen::MatrixBase<DerivedDP>& dp, const Eigen::MatrixBase<DerivedDT0>& dt0, const Eigen::MatrixBase<DerivedDT1>& dt1, const Eigen::MatrixBase<DerivedDT2>& dt2, const Eigen::MatrixBase<DerivedCoords>& coords);¶

Compute the relative velocity of the point to the triangle.

Parameters:¶

const Eigen::MatrixBase<DerivedDP> &dp¶: Velocity of the point
const Eigen::MatrixBase<DerivedDT0> &dt0¶: Velocity of the first vertex of the triangle
const Eigen::MatrixBase<DerivedDT1> &dt1¶: Velocity of the second vertex of the triangle
const Eigen::MatrixBase<DerivedDT2> &dt2¶: Velocity of the third vertex of the triangle
const Eigen::MatrixBase<DerivedCoords> &coords¶: Baricentric coordinates of the closest point on the triangle

Returns:¶

The relative velocity of the point to the triangle

Relative Velocity as Multiplier Matricies¶

Warning

doxygenfunction: Unable to resolve function “ipc::point_point_relative_velocity_jacobian” with arguments “None”. Candidate function could not be parsed. Parsing error is Error when parsing function declaration. If the function has no return type: Error in declarator or parameters-and-qualifiers Invalid C++ declaration: Expecting “(” in parameters-and-qualifiers. [error at 54] template<typename T = double> IPC_TOOLKIT_HOST_DEVICE MatrixMax< T, 3, 6 > point_point_relative_velocity_jacobian (const int dim) ——————————————————^ If the function has a return type: Error in declarator or parameters-and-qualifiers If pointer to member declarator: Invalid C++ declaration: Expected ‘::’ in pointer to member (function). [error at 75] template<typename T = double> IPC_TOOLKIT_HOST_DEVICE MatrixMax< T, 3, 6 > point_point_relative_velocity_jacobian (const int dim) —————————————————————————^ If declarator-id: Invalid C++ declaration: Expecting “(” in parameters-and-qualifiers. [error at 75] template<typename T = double> IPC_TOOLKIT_HOST_DEVICE MatrixMax< T, 3, 6 > point_point_relative_velocity_jacobian (const int dim) —————————————————————————^

Warning

doxygenfunction: Unable to resolve function “ipc::point_edge_relative_velocity_jacobian” with arguments “None”. Candidate function could not be parsed. Parsing error is Error when parsing function declaration. If the function has no return type: Error in declarator or parameters-and-qualifiers Invalid C++ declaration: Expecting “(” in parameters-and-qualifiers. [error at 54] template<typename T = double> IPC_TOOLKIT_HOST_DEVICE MatrixMax< T, 3, 9 > point_edge_relative_velocity_jacobian (const int dim, const T alpha) ——————————————————^ If the function has a return type: Error in declarator or parameters-and-qualifiers If pointer to member declarator: Invalid C++ declaration: Expected ‘::’ in pointer to member (function). [error at 75] template<typename T = double> IPC_TOOLKIT_HOST_DEVICE MatrixMax< T, 3, 9 > point_edge_relative_velocity_jacobian (const int dim, const T alpha) —————————————————————————^ If declarator-id: Invalid C++ declaration: Expecting “(” in parameters-and-qualifiers. [error at 75] template<typename T = double> IPC_TOOLKIT_HOST_DEVICE MatrixMax< T, 3, 9 > point_edge_relative_velocity_jacobian (const int dim, const T alpha) —————————————————————————^

template <typename DerivedCoords> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::edge_edge_relative_velocity_jacobian( const Eigen::MatrixBase<DerivedCoords>& coords);¶

Compute du/dx where u is the relative velocity of two edges.

Parameters:¶

const Eigen::MatrixBase<DerivedCoords> &coords¶: Two parametric coordinates of the closest points on the edges

Returns:¶

The relative velocity Jacobian du/dx

template <typename DerivedCoords> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_triangle_relative_velocity_jacobian( const Eigen::MatrixBase<DerivedCoords>& coords);¶

Compute du/dx where u is the relative velocity of a point and a triangle.

Parameters:¶

const Eigen::MatrixBase<DerivedCoords> &coords¶: Barycentric coordinates of the closest point on the triangle

Returns:¶

The relative velocity Jacobian du/dx

Relative Velocity Matrix Jacobians¶

Warning

doxygenfunction: Unable to resolve function “ipc::point_point_relative_velocity_dx_dbeta” with arguments “None”. Candidate function could not be parsed. Parsing error is Error when parsing function declaration. If the function has no return type: Error in declarator or parameters-and-qualifiers Invalid C++ declaration: Expecting “(” in parameters-and-qualifiers. [error at 54] template<typename T = double> IPC_TOOLKIT_HOST_DEVICE VectorMax< T, 18 > point_point_relative_velocity_dx_dbeta (const int dim) ——————————————————^ If the function has a return type: Error in declarator or parameters-and-qualifiers If pointer to member declarator: Invalid C++ declaration: Expected ‘::’ in pointer to member (function). [error at 73] template<typename T = double> IPC_TOOLKIT_HOST_DEVICE VectorMax< T, 18 > point_point_relative_velocity_dx_dbeta (const int dim) ————————————————————————-^ If declarator-id: Invalid C++ declaration: Expecting “(” in parameters-and-qualifiers. [error at 73] template<typename T = double> IPC_TOOLKIT_HOST_DEVICE VectorMax< T, 18 > point_point_relative_velocity_dx_dbeta (const int dim) ————————————————————————-^

Warning

doxygenfunction: Unable to resolve function “ipc::point_edge_relative_velocity_dx_dbeta” with arguments “None”. Candidate function could not be parsed. Parsing error is Error when parsing function declaration. If the function has no return type: Error in declarator or parameters-and-qualifiers Invalid C++ declaration: Expecting “(” in parameters-and-qualifiers. [error at 54] template<typename T = double> IPC_TOOLKIT_HOST_DEVICE VectorMax< T, 27 > point_edge_relative_velocity_dx_dbeta (const int dim, const T alpha) ——————————————————^ If the function has a return type: Error in declarator or parameters-and-qualifiers If pointer to member declarator: Invalid C++ declaration: Expected ‘::’ in pointer to member (function). [error at 73] template<typename T = double> IPC_TOOLKIT_HOST_DEVICE VectorMax< T, 27 > point_edge_relative_velocity_dx_dbeta (const int dim, const T alpha) ————————————————————————-^ If declarator-id: Invalid C++ declaration: Expecting “(” in parameters-and-qualifiers. [error at 73] template<typename T = double> IPC_TOOLKIT_HOST_DEVICE VectorMax< T, 27 > point_edge_relative_velocity_dx_dbeta (const int dim, const T alpha) ————————————————————————-^

template <typename DerivedCoords> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::edge_edge_relative_velocity_dx_dbeta( const Eigen::MatrixBase<DerivedCoords>& coords);¶

Compute d²u/dxdβ where u is the relative velocity of two edges.

Γ(β₁,β₂) = [(1-β₁)I, β₁I, (β₂-1)I, -β₂I] (3 × 12) ∂Γ/∂β₁ = [-I, I, 0, 0] ∂Γ/∂β₂ = [ 0, 0, I,-I]

Stored as [vec(∂Γ/∂β₁) | vec(∂Γ/∂β₂)] in column-major order (3rd-order convention). Result shape: (36, 2). For a (3 × 12) matrix M, element M(r,c) maps to vec index c·3 + r.

Parameters:¶

const Eigen::MatrixBase<DerivedCoords> &coords¶: Two parametric coordinates of the closest points on the edges

Returns:¶

The vectorized tensor of d²u/dxdβ

template <typename DerivedCoords> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_triangle_relative_velocity_dx_dbeta( const Eigen::MatrixBase<DerivedCoords>& coords);¶

Compute d²u/dxdβ where u is the relative velocity of a point and a triangle.

Γ(β₁,β₂) = [I, (β₁+β₂-1)I, -β₁I, -β₂I] (3 × 12) ∂Γ/∂β₁ = [0, I, -I, 0] ∂Γ/∂β₂ = [0, I, 0,-I]

Stored as [vec(∂Γ/∂β₁) | vec(∂Γ/∂β₂)] in column-major order (3rd-order convention). Result shape: (36, 2). For a (3 × 12) matrix M, element M(r,c) maps to vec index c·3 + r.

Parameters:¶

const Eigen::MatrixBase<DerivedCoords> &coords¶: Baricentric coordinates of the closest point on the triangle

Returns:¶

The vectorized tensor of d²u/dxdβ

Closet Points¶

template <typename DerivedP, typename DerivedE0, typename DerivedE1> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_edge_closest_point( const Eigen::MatrixBase<DerivedP>& p, const Eigen::MatrixBase<DerivedE0>& e0, const Eigen::MatrixBase<DerivedE1>& e1);¶

Compute the baricentric coordinate of the closest point on the edge.

Parameters:¶

const Eigen::MatrixBase<DerivedP> &p¶: Point
const Eigen::MatrixBase<DerivedE0> &e0¶: First edge point
const Eigen::MatrixBase<DerivedE1> &e1¶: Second edge point

Returns:¶

barycentric coordinates of the closest point

template <typename DerivedEA0, typename DerivedEA1, typename DerivedEB0, typename DerivedEB1> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::edge_edge_closest_point( const Eigen::MatrixBase<DerivedEA0>& ea0, const Eigen::MatrixBase<DerivedEA1>& ea1, const Eigen::MatrixBase<DerivedEB0>& eb0, const Eigen::MatrixBase<DerivedEB1>& eb1);¶

Compute the barycentric coordinates of the closest points between two edges.

Accepts any 3D Eigen expression; the scalar type is deduced.

Parameters:¶

const Eigen::MatrixBase<DerivedEA0> &ea0¶: First point of the first edge
const Eigen::MatrixBase<DerivedEA1> &ea1¶: Second point of the first edge
const Eigen::MatrixBase<DerivedEB0> &eb0¶: First point of the second edge
const Eigen::MatrixBase<DerivedEB1> &eb1¶: Second point of the second edge

Returns:¶

Barycentric coordinates of the closest points

template <typename DerivedP, typename DerivedT0, typename DerivedT1, typename DerivedT2> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_triangle_closest_point( const Eigen::MatrixBase<DerivedP>& p, const Eigen::MatrixBase<DerivedT0>& t0, const Eigen::MatrixBase<DerivedT1>& t1, const Eigen::MatrixBase<DerivedT2>& t2);¶

Compute the barycentric coordinates of the closest point on the triangle.

Accepts any 3D Eigen expression; the scalar type is deduced.

Parameters:¶

const Eigen::MatrixBase<DerivedP> &p¶: Point
const Eigen::MatrixBase<DerivedT0> &t0¶: Triangle’s first vertex
const Eigen::MatrixBase<DerivedT1> &t1¶: Triangle’s second vertex
const Eigen::MatrixBase<DerivedT2> &t2¶: Triangle’s third vertex

Returns:¶

Barycentric coordinates of the closest point

Closet Points Jacobians¶

template <typename DerivedP, typename DerivedE0, typename DerivedE1> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_edge_closest_point_jacobian( const Eigen::MatrixBase<DerivedP>& p, const Eigen::MatrixBase<DerivedE0>& e0, const Eigen::MatrixBase<DerivedE1>& e1);¶

Compute the Jacobian of the closest point on the edge.

Parameters:¶

const Eigen::MatrixBase<DerivedP> &p¶: Point
const Eigen::MatrixBase<DerivedE0> &e0¶: First edge point
const Eigen::MatrixBase<DerivedE1> &e1¶: Second edge point

Returns:¶

Jacobian of the closest point

template <typename DerivedEA0, typename DerivedEA1, typename DerivedEB0, typename DerivedEB1> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::edge_edge_closest_point_jacobian( const Eigen::MatrixBase<DerivedEA0>& ea0, const Eigen::MatrixBase<DerivedEA1>& ea1, const Eigen::MatrixBase<DerivedEB0>& eb0, const Eigen::MatrixBase<DerivedEB1>& eb1);¶

Compute the Jacobian of the closest points between two edges.

Accepts any 3D Eigen expression; the scalar type is deduced.

Parameters:¶

const Eigen::MatrixBase<DerivedEA0> &ea0¶: First point of the first edge
const Eigen::MatrixBase<DerivedEA1> &ea1¶: Second point of the first edge
const Eigen::MatrixBase<DerivedEB0> &eb0¶: First point of the second edge
const Eigen::MatrixBase<DerivedEB1> &eb1¶: Second point of the second edge

Returns:¶

Jacobian of the closest points

template <typename DerivedP, typename DerivedT0, typename DerivedT1, typename DerivedT2> inline IPC_TOOLKIT_HOST_DEVICE auto ipc::point_triangle_closest_point_jacobian( const Eigen::MatrixBase<DerivedP>& p, const Eigen::MatrixBase<DerivedT0>& t0, const Eigen::MatrixBase<DerivedT1>& t1, const Eigen::MatrixBase<DerivedT2>& t2);¶

Compute the Jacobian of the closest point on the triangle.

Accepts any 3D Eigen expression; the scalar type is deduced.

Parameters:¶

const Eigen::MatrixBase<DerivedP> &p¶: Point
const Eigen::MatrixBase<DerivedT0> &t0¶: Triangle’s first vertex
const Eigen::MatrixBase<DerivedT1> &t1¶: Triangle’s second vertex
const Eigen::MatrixBase<DerivedT2> &t2¶: Triangle’s third vertex

Returns:¶

Jacobian of the closest point